Difference Between Horizontal and Vertical Asymptote

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small.

1. What is the horizontal asymptote?
2. What is a vertical asymptote?
3. How do you tell the difference between a vertical asymptote and a hole?
4. How do you tell if there is a horizontal asymptote?
5. How many horizontal asymptotes can a function have?
6. Where do vertical asymptotes occur?
7. How do you find vertical asymptotes and horizontal holes?
8. Are holes undefined?
9. How can you tell if a graph is a rational function?

What is the horizontal asymptote?

Horizontal asymptotes are horizontal lines the graph approaches. ... If the degree (the largest exponent) of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x-axis (y = 0).

What is a vertical asymptote?

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.)

How do you tell the difference between a vertical asymptote and a hole?

Earlier, you were asked how asymptotes are different than holes. Holes occur when factors from the numerator and the denominator cancel. When a factor in the denominator does not cancel, it produces a vertical asymptote.

How do you tell if there is a horizontal asymptote?

If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote. If the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote.

How many horizontal asymptotes can a function have?

A function can have at most two different horizontal asymptotes.

Where do vertical asymptotes occur?

In general, a vertical asymptote occurs in a rational function at any value of x for which the denominator is equal to 0, but for which the numerator is not equal to 0.

How do you find vertical asymptotes and horizontal holes?

Set each factor in the denominator equal to zero and solve for the variable. If this factor does not appear in the numerator, then it is a vertical asymptote of the equation. If it does appear in the numerator, then it is a hole in the equation.

Are holes undefined?

Holes and Rational Functions

A hole on a graph looks like a hollow circle. ... As you can see, f(−12) is undefined because it makes the denominator of the rational part of the function zero which makes the whole function undefined.

How can you tell if a graph is a rational function?

A rational function will be zero at a particular value of x only if the numerator is zero at that x and the denominator isn't zero at that x . In other words, to determine if a rational function is ever zero all that we need to do is set the numerator equal to zero and solve.